# Nuclear Forward Scattering

## Dynamical theory / Multiple scattering

### Collective coherent scattering process

If a g-quantum impinges on a target composed of identical nuclei, it can interact with any of them. When the scattering process is elastic, it is impossible to determine which nucleus has been excited, therefore all possible paths of scattering have to be taken into account.

The excitation produced by the g-ray is therefore de-localised (nuclear exciton), and the scattering process has a collective character. The waves re-radiated by different nuclei will in general interfere with one another (summation of amplitudes instead of intensities!).

In the case of a disordered nuclear system only the waves scattered in direction of the incident beam (forward direction) will keep a fixed phase relation to one another, leading to a spatially coherent scattering process.

The g-quantum can therefore either pass the target without interacting or be coherently scattered in forward direction, and the transmitted wave-field is a coherent superposition of incident and forward scattered wave-fields.

 Etr = Ei + Efwd.

### The influence of the thickness of the target / Multiple scattering

The transmitted wave field through an infinitesimal thin layer (with thickness dz much smaller than the absorption length and negligible electronic absorption) can be calculated by summing up the waves reradiated by the individual nuclei which were excited by the primary beam

 Etr = Ei + Efwd = Ei - i Ei l a0 nmb ·dz  .
 with a0 the elastic scattering amplitude (8) and nmb the concentration of resonant nuclei

Taking the transmitted radiation through one layer as incident beam for the next layer the transmission through a slab of arbitrary thickness d is determined by a differential equation of the field amplitude

 dEtr (z) = - i Etr(z) l a0 nmb ·dz  .
In other words, the coherent scattering by nuclei upstream of the regarded layer take part in the formation of this field and thus the effect of multiple scattering is taken into account.
Integrating the differential equation leads to the following
 Etr = Ei · exp[- i l a0 nmb d] = Ei · exp[ l Im(a0) nmb d] · exp[-i l Re(a0) nmb d] attenuation of amplitude · shift of phase
While the incident radiation travels through the target each component is modified due to forward scattering:
 At resonance energy the imaginary part of the scattering amplitude has a minimum, whereas the real part is zero, which means no phase shift and highest attenuation. Far off resonance the imaginary part is close to zero, there is only very little attenuation, but a large phase shift (depending on the thickness of the target).

For a conventional Mössbauer experiment, where we are not sensitive to interferences between the different harmonics, the transmitted intensity is determined only by the square of the transmitted amplitude
 Itr(d,w) = |Etr|2
 =
 I0(w) exp[2 l  Im(a0(w)) nmb d]
 =
 I0(w) exp[- sR(w)nmb d]
with the nuclear absorption cross section sR determined by the imaginary part of the forward scattering amplitude (in accordance with the optical theorem).

The real part of the scattering amplitude, which contains the information about the phase shift, is lost in conventional experiments.

Based on the PhD thesis
of Alessandro Barla, Herdecke 2001
and Hanne Grünsteudel, Lübeck 1998